Gibbs measures and phase transitions in various one-dimensional models

Abstract

In the thesis, limiting Gibbs measures of some one dimensional models are investigated and various criterions for the uniqueness of limiting Gibbs states are considered. The criterion for models with unique ground state formulated in terms of percolation theory is presented and some applications of this criterion are discussed. A one-dimensional long range Widom-Rowlinson model with periodic and biased particle activities is explored. It is shown that if the spin interactions are sufficiently large versus particle activities then the Widom-Rowlinson model does not exhibit a phase transition at low temperatures. Finally, an interdisciplinary approach is followed. A financial application of the theory of phase transition is considered by applying the Ising model to understand the role of herd behavior on stock market crashes. Accordingly, model suggests a criteria to detect the existence of herd behavior in financial markets under certain assumptions.